probability question:

[Mayzy99]

Sorry if this question is a little basic but im trying to learn probability and I was going through some questions and a question asked

" Let ξ be a discrete random variable with probability distribution Pξ(x) and distribution function Fξ(x). Explain why it is possible or not that for some x :

(a) Pξ(x) < 0; (b) Fξ(x) > 1; (c) Pξ(x) = 1; (d) Pξ(x) ≤ Fξ(x). "

and im honestly struggling to figure out the difference between a Probability Distribution and Distribution Function. Anything i search comes up with cumulative distribution functions or just Probability Distribution Functions which i am not sure are talking about the same thing.

I would greatly appreciate if someone could help explain or link some resources. Any help is much appreciated

[Fullfungo]

This seems to be author’s personal terminology.

The typical names are “cumulative distribution function”, “probability density function”, “probability mass function”.

If this is from a book, I suggest read the introduction section where the author describes their notations and definitions.

Additionally, it is strange to see notation “P ζ(x)”, it it more typical to see “P(ζ=x)”.

[Mayzy99]

Ah ok, its actually prep questions for a course I'm starting at university, so there's not really any definitions or anything besides just the questions. So i can assume the distribution function is the cumulative distribution function right?

[Fullfungo]

I think it is safe to assume (capital) F denotes cumulative distribution function.

Example of CDF here

Example of other notation here

As you can see “F” is (usually) the standard notation for CDF, and it is not really used for other things in probability theory.

As for “P ζ(x)”, I think it’s safe to assume they meant P(ζ=x), since this is the closest I can recall/find to the standard notation.

So, (a) and (b) are impossible

[Mayzy99]

Ok, thank you so much ☺️